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A347465 Numbers whose multiset of prime indices has alternating product > 1.

%I A347465 #9 Nov 03 2021 10:47:23
%S A347465 3,5,7,11,12,13,17,19,20,23,27,28,29,30,31,37,41,42,43,44,45,47,48,52,
%T A347465 53,59,61,63,66,67,68,70,71,73,75,76,78,79,80,83,89,92,97,99,101,102,
%U A347465 103,105,107,108,109,110,112,113,114,116,117,120,124,125,127
%N A347465 Numbers whose multiset of prime indices has alternating product > 1.
%C A347465 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A347465 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347465 All terms have odd bigomega (A001222).
%C A347465 Also Heinz numbers integer partitions with reverse-alternating product > 1.
%e A347465 The terms and their prime indices begin:
%e A347465       3: {2}         37: {12}            68: {1,1,7}
%e A347465       5: {3}         41: {13}            70: {1,3,4}
%e A347465       7: {4}         42: {1,2,4}         71: {20}
%e A347465      11: {5}         43: {14}            73: {21}
%e A347465      12: {1,1,2}     44: {1,1,5}         75: {2,3,3}
%e A347465      13: {6}         45: {2,2,3}         76: {1,1,8}
%e A347465      17: {7}         47: {15}            78: {1,2,6}
%e A347465      19: {8}         48: {1,1,1,1,2}     79: {22}
%e A347465      20: {1,1,3}     52: {1,1,6}         80: {1,1,1,1,3}
%e A347465      23: {9}         53: {16}            83: {23}
%e A347465      27: {2,2,2}     59: {17}            89: {24}
%e A347465      28: {1,1,4}     61: {18}            92: {1,1,9}
%e A347465      29: {10}        63: {2,2,4}         97: {25}
%e A347465      30: {1,2,3}     66: {1,2,5}         99: {2,2,5}
%e A347465      31: {11}        67: {19}           101: {26}
%t A347465 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347465 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347465 Select[Range[100],altprod[primeMS[#]]>1&]
%Y A347465 The squarefree case is A030059 without 2.
%Y A347465 The reverse version is A028983, counted by A119620.
%Y A347465 The opposite version (< 1 instead of > 1) is A119899.
%Y A347465 Factorizations of this type are counted by A339890, reverse A347705.
%Y A347465 The weak version (>= 1 instead of > 1) is A344609.
%Y A347465 Partitions of this type are counted by A347449, reverse A347448.
%Y A347465 The complement is A347450, counted by A339846 or A347443.
%Y A347465 Allowing any integer reverse-alternating product gives A347454.
%Y A347465 Allowing any integer alternating product gives A347457.
%Y A347465 A335433 ranks inseparable partitions, complement A335448.
%Y A347465 A347446 counts partitions with integer alternating product, reverse A347445.
%Y A347465 Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.
%K A347465 nonn
%O A347465 1,1
%A A347465 _Gus Wiseman_, Sep 27 2021